Precalculus Examples

$| \frac{x - 1}{3} | < 2$

Step 1

Write $| \frac{x - 1}{3} | < 2$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{x - 1}{3} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $3$ .

$\frac{x - 1}{3} \cdot 3 \geq 0 \cdot 3$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{x - 1}{3} \cdot 3 \geq 0 \cdot 3$

Step 1.2.2.1.1.2

Rewrite the expression.

$x - 1 \geq 0 \cdot 3$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $3$ .

$x - 1 \geq 0$

Step 1.2.3

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 1.3

In the piece where $\frac{x - 1}{3}$ is non-negative, remove the absolute value.

$\frac{x - 1}{3} < 2$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{x - 1}{3} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $3$ .

$\frac{x - 1}{3} \cdot 3 < 0 \cdot 3$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1.1

Cancel the common factor.

$\frac{x - 1}{3} \cdot 3 < 0 \cdot 3$

Step 1.5.2.1.1.2

Rewrite the expression.

$x - 1 < 0 \cdot 3$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $3$ .

$x - 1 < 0$

Step 1.5.3

Add $1$ to both sides of the inequality.

$x < 1$

Step 1.6

In the piece where $\frac{x - 1}{3}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{x - 1}{3} < 2$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{x - 1}{3} < 2 & x \geq 1 \\ - \frac{x - 1}{3} < 2 & x < 1 \end{cases}$

Step 2

Solve $\frac{x - 1}{3} < 2$ when $x \geq 1$ .

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Step 2.1

Solve $\frac{x - 1}{3} < 2$ for $x$ .

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Step 2.1.1

Multiply both sides by $3$ .

$\frac{x - 1}{3} \cdot 3 < 2 \cdot 3$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the left side.

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Step 2.1.2.1.1

Cancel the common factor of $3$ .

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Step 2.1.2.1.1.1

Cancel the common factor.

$\frac{x - 1}{3} \cdot 3 < 2 \cdot 3$

Step 2.1.2.1.1.2

Rewrite the expression.

$x - 1 < 2 \cdot 3$

Step 2.1.2.2

Simplify the right side.

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Step 2.1.2.2.1

Multiply $2$ by $3$ .

$x - 1 < 6$

Step 2.1.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.3.1

Add $1$ to both sides of the inequality.

$x < 6 + 1$

Step 2.1.3.2

Add $6$ and $1$ .

$x < 7$

Step 2.2

Find the intersection of $x < 7$ and $x \geq 1$ .

$1 \leq x < 7$

Step 3

Solve $- \frac{x - 1}{3} < 2$ when $x < 1$ .

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Step 3.1

Solve $- \frac{x - 1}{3} < 2$ for $x$ .

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Step 3.1.1

Divide each term in $- \frac{x - 1}{3} < 2$ by $- 1$ and simplify.

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Step 3.1.1.1

Divide each term in $- \frac{x - 1}{3} < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 1}{3}}{- 1} > \frac{2}{- 1}$

Step 3.1.1.2

Simplify the left side.

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Step 3.1.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 1}{3}}{1} > \frac{2}{- 1}$

Step 3.1.1.2.2

Divide $\frac{x - 1}{3}$ by $1$ .

$\frac{x - 1}{3} > \frac{2}{- 1}$

Step 3.1.1.3

Simplify the right side.

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Step 3.1.1.3.1

Divide $2$ by $- 1$ .

$\frac{x - 1}{3} > - 2$

Step 3.1.2

Multiply both sides by $3$ .

$\frac{x - 1}{3} \cdot 3 > - 2 \cdot 3$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Simplify the left side.

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Step 3.1.3.1.1

Cancel the common factor of $3$ .

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Step 3.1.3.1.1.1

Cancel the common factor.

$\frac{x - 1}{3} \cdot 3 > - 2 \cdot 3$

Step 3.1.3.1.1.2

Rewrite the expression.

$x - 1 > - 2 \cdot 3$

Step 3.1.3.2

Simplify the right side.

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Step 3.1.3.2.1

Multiply $- 2$ by $3$ .

$x - 1 > - 6$

Step 3.1.4

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.4.1

Add $1$ to both sides of the inequality.

$x > - 6 + 1$

Step 3.1.4.2

Add $- 6$ and $1$ .

$x > - 5$

Step 3.2

Find the intersection of $x > - 5$ and $x < 1$ .

$- 5 < x < 1$

Step 4

Find the union of the solutions.

$- 5 < x < 7$

Step 5

Convert the inequality to interval notation.

$(- 5, 7)$

Step 6